A326573 Number of connected antichains of subsets of {1..n}, all having different sums.

1, 1, 1, 5, 59, 2689, 787382

Offset: 0

Author

Gus Wiseman, Jul 18 2019

Keywords

• nonn
• more

Comments

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

Examples

			The a(3) = 5 antichains:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 59 antichains:
  {1234}  {12}{134}   {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}
          {12}{234}   {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}
          {13}{124}   {12}{13}{34}   {12}{13}{23}{24}
          {13}{234}   {12}{14}{34}   {12}{13}{23}{34}
          {14}{123}   {12}{23}{24}   {12}{13}{24}{34}
          {14}{234}   {12}{23}{34}   {12}{14}{24}{34}
          {23}{124}   {12}{24}{34}   {12}{23}{24}{34}
          {23}{134}   {13}{14}{24}   {13}{14}{24}{34}
          {24}{134}   {13}{23}{24}   {13}{23}{24}{34}
          {34}{123}   {13}{23}{34}   {12}{13}{14}{234}
          {123}{124}  {13}{24}{34}   {12}{23}{24}{134}
          {123}{134}  {14}{24}{34}   {123}{124}{134}{234}
          {123}{234}  {12}{13}{234}
          {124}{134}  {12}{14}{234}
          {124}{234}  {12}{23}{134}
          {134}{234}  {12}{24}{134}
                      {13}{14}{234}
                      {13}{23}{124}
                      {14}{34}{123}
                      {23}{24}{134}
                      {12}{134}{234}
                      {13}{124}{234}
                      {14}{123}{234}
                      {23}{124}{134}
                      {123}{124}{134}
                      {123}{124}{234}
                      {123}{134}{234}
                      {124}{134}{234}
		

Crossrefs

Antichain covers are A006126.

Connected antichains are A048143.

Set partitions with different block-sums are A275780.

MM-numbers of multiset partitions with different part-sums are A326535.

Antichain covers with equal edge-sums are A326566.

The non-connected case is A326572.

Cf. A000372, A293510, A307249, A321469, A323818, A326519, A326565, A326569, A326570, A326571.